When I think it's low odds of a hurricane damaging the Lychee trees this year, I will pull out the knife, constrictor cords, and other instruments of destruction to put a hurting on a few lychee tree limbs as the Sept. to Oct. damaged tree limbs seem to produce more fruit the following year in my yard... I think it's the feeling of possible death entering the tree and thus reproduction (fruiting) becomes more of a priority. Your mileage may vary & I would not damage the main trunk from the roots up to the first limb union. However if 3 limbs branch off 1 trunk, it's easy to try this on at least one limb division for next year to check if it works for you.... And if you like the results, the following year you could damage 2 of the 3 branches etc. etc. etc. if it works for you.
If your tree is bearing good then of course do not cut it, keep it as is.